Remarks on “Calculation of the quarkonium spectrum andmb, mcto orderαs4”

Abstract

In a recent paper, we included two-loop, relativistic one-loop, and second-order relativistic tree level corrections, plus leading nonperturbative contributions, to obtain a calculation of the lower states in the heavy quarkonium spectrum correct up to, and including, $O({\ensuremath{\alpha}}_{s}^{4})$ and leading ${\ensuremath{\Lambda}}^{4}{/m}^{4}$ terms. The results were obtained with, in particular, the value of the two-loop static coefficient due to Peter; this has been recently challenged by Schr\"oder. In our previous paper we used Peter's result; in the present one we now give results with Schr\"oder's, as this is likely to be the correct one. The variation is slight as the value of ${b}_{1}$ is only one among the various $O({\ensuremath{\alpha}}_{s}^{4})$ contributions. With Schr\"oder's expression we now have ${m}_{b}{=5001}_{\ensuremath{-}66}^{+104} \mathrm{MeV},$ ${m}_{b}({m}_{b}^{2}{)=4454}_{\ensuremath{-}29}^{+45} \mathrm{MeV},$ ${m}_{c}{=1866}_{\ensuremath{-}133}^{+215} \mathrm{MeV},$ ${m}_{c}({m}_{c}^{2}{)=1542}_{\ensuremath{-}104}^{+163} \mathrm{MeV}.$ Moreover, $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Upsilon}}{e}^{+}{e}^{\ensuremath{-}})=1.07\ifmmode\pm\else\textpm\fi{}0.28 \mathrm{keV}(\mathrm{expt}=1.320\ifmmode\pm\else\textpm\fi{}0.04 \mathrm{keV})$ and the hyperfine splitting is predicted to be $M(\ensuremath{\Upsilon})\ensuremath{-}M(\ensuremath{\eta}{)=47}_{\ensuremath{-}13}^{+15} \mathrm{MeV}.$